1　Compressible Flow and Non-linear Wave Equations
　1.1　Euler's Equations
　1.2　Irrotational Flow and the Nonlinear Wave Equation
　1.3　The Equation of Variations and the Acoustical Metric
　1.4　The Fundamental Variations
2　The Basic Geometric Construction
　2.1　Null Foliation Associated with the Acoustical Metric
　　2.1.1　Galilean Spacetime
　　2.1.2　Null Foliation and Acoustical Coordinates
　2.2　A Geometric Interpretation for Function H
3　The Acoustical Structure Equations
　3.1　The Acoustical Structure Equations
　3.2　The Derivatives of the Rectangular Components of L and T
4　The Acoustical Curvature
　4.1　Expressions for Curvature Tensor 1　Compressible Flow and Non-linear Wave Equations
　1.1　Euler's Equations
　1.2　Irrotational Flow and the Nonlinear Wave Equation
　1.3　The Equation of Variations and the Acoustical Metric
　1.4　The Fundamental Variations
2　The Basic Geometric Construction
　2.1　Null Foliation Associated with the Acoustical Metric
　　2.1.1　Galilean Spacetime
　　2.1.2　Null Foliation and Acoustical Coordinates
　2.2　A Geometric Interpretation for Function H
3　The Acoustical Structure Equations
　3.1　The Acoustical Structure Equations
　3.2　The Derivatives of the Rectangular Components of L and T
4　The Acoustical Curvature
　4.1　Expressions for Curvature Tensor
　4.2　Regularity for the Acoustical Structure Equations as μ → 0
　4.3　A Remark
5　The Fundamental Energy Estimate
　5.1　Bootstrap Assumptions. Statement of the Theorem
　5.2　The Multiplier Fields K0 and K1. The Associated Energy-Momentum Density Vectorfields
　5.3　The Error Integrals
　5.4　The Estimates for the Error Integrals
　5.5　Treatment of the Integral Inequalities Depending on t and u.
　　Completion of the Proof
6　Construction of Commutation Vectorfields
　6.1　Commutation Vectorfields and Their Deformation Tensors
　6.2　Preliminary Estimates for the Deformation Tensors
7　Outline of the Derived Estimates of Each Order
　7.1　The Inhomogeneous Wave Equations for the Higher Order Variations. The Recursion Formula for the Source Functions
　7.2　The First Term in ρn
　7.3　The Estimates of the Contribution of the First Term in ρn to the Error Integrals
8　Regularization of the Propagation Equation for □trX. Estimates for the Top Order Angular Derivatives of X
　8.1　Preliminary
　　8.1.1　Regularization of The Propagation Equation
　　8.1.2　Propagation Equations for Higher Order Angular Derivatives
　　8.1.3　Elliptic Theory on St,u
　　8.1.4　Preliminary Estimates for the Solutions of the Propagation Equations
　8.2　Crucial Lemmas Concerning the Behavior of μ
　8.3　The Actual Estimates for the Solutions of the Propagation Equations
9　 Regularization of the Propagation Equation for □μ.
　Estimates for the Top Order Spatial Derivatives of μ
　9.1　Regularization of the Propagation Equation
　9.2　Propagation Equations for the Higher Order Spatial Derivatives
　9.3　Elliptic Theory on St,u
　9.4　The Estimates for the Solutions of the Propagation Equations
10　Control of the Angular Derivatives of the First Derivatives of
　the xi. Assumptions and Estimates in Regard to X
　10.1 Preliminary
　10.2 Estimates for yi
　　10.2.1 L∞ Estimates for Rik ... .Ri1yj
　　10.2.2 L2 Estimates for Rik... Pi1yj
　10.3 Bounds for the quantities Ql and Pl
　　10.3.1 Estimates for Ql
　　10.3.2 Estimates for Pl
11　Control of the Spatial Derivatives of the First Derivatives of
　the xi. Assumptions and Estimates in Regard to μ
　11.1 Estimates for TTi
　　11.1.1 Basic Lemmas
　　11.1.2 L∞ Estimates for TTi
　　11.1.3 L2 Estimates for TTi
　11.2 Bounds for Quantities Q'm,l and P'm,l
　　11.2.1 Bounds for Q'm,l
　　11.2.2 Bounds for P'm,l
12　Recovery of the Acoustical Assumptions
　Estimates for Up to the Next to the Top Order Angular
　Derivatives of X and Spatial Derivatives of μ
　12.1　Estimates for λi, y', yi and r. Establishing the Hypothesis H0
　12.2　The Coercivity Hypothesis H1, H2 and H2'. Estimates for X'
　12.3　Estimates for Higher Order Derivatives of X' and μ
13　Derivation of the Basic Properties of μ
14　The Error Estimates Involving the Top Order Spatial
　Derivatives of the Acoustical Entities
　14.1 The Error Terms Involving the Top Order Spatial
　　Derivatives of the Acoustical Entities
　14.2 The Borderline Error Integrals
　14.3 Assumption J
　14.4 The Borderline Estimates Associated to K0
　　14.4.1 Estimates for the Contribution of (14.56)
　　14.4.2 Estimates for the Contribution of (14.57)
　14.5 The Borderline Estimates Associated to K1
　　14.5.1 Estimates for the Contribution of (14.56)
　　14.5.2 Estimates for the Contribution of (14.57)
15　The Top Order Energy Estimates
　15.1 Estimates Associated to K1
　15.2 Estimates Associated to K0
16　The Descent Scheme
17　The Isoperimetric Inequality. Recovery of Assumption J.
　 Recovery of the Bootstrap Assumption. Proof of the Main
　 Theorem
　17.1 Recovery of J--Preliminary
　17.2 The Isoperimetric Inequality
　17.3 Recovery of J--Completion
　17.4 Recovery of the Final Bootstrap Assumption
　17.5 Completion of the Proof of the Main Theorem
18　Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolution
19　The Structure of the Boundary of the Domain of the Maximal Solution
　19.1 Nature of Singular Hypersurface in Acoustical Differential Structure
　　19.1.1 Preliminary
　　19.1.2 Intrinsic View Point
　　19.1.3 Invariant Curves
　　19.1.4 Extrinsic View Point
　19.2 The Trichotomy Theorem for Past Null Geodesics Ending at Singular Boundary
　　19.2.1 Hamiltonian Flow
　　19.2.2 Asymptotic Behavior
　19.3 Transformation of Coordinates
　19.4 How H Looks Like in Rectangular Coordinates in Galilean Spacetime
References